An Extremal Eigenvalue Problem for a Two-Phase Conductor in a Ball

The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materials. PNLDE 31. Birkhäuser, Basel, 1997) go a long way in showing, in general, that problems of optimal design may not admit solutions if microstructural designs are excluded from consideration. Therefore, assuming, tactilely, that the problem of minimizing the first eigenvalue of a two-phase conducting material with the conducting phases to be distributed in a fixed proportion in a given domain has no true solution in general domains, Cox and Lipton only study conditions for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech. Anal. 136:101–117, 1996). Although, the problem in one dimension has a solution (cf. Kre˘ın in AMS Transl. Ser. 2(1):163–187, 1955) and, in higher dimensions, the problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino et al. in Nonlinear Anal. TMA 13(2):185–220, 1989), these existence results have been regarded so far as being exceptional owing to complete symmetry. It is still not clear why the same problem in domains with partial symmetry should fail to have a solution which does not develop microstructure and respecting the symmetry of the domain.We hope to revive interest in this question by giving a new proof of the result in a ball using a simpler symmetrization result from Alvino and Trombetti

Spectral asymptotics of the Helmholtz model in fluid-solid structures

Artículo de publicación ISIA model representing the vibrations of a coupled fluid-solid structure is considered. This structure consists of a tube bundle immersed in a slightly compressible fluid. Assuming periodic distribution of tubes, this article describes the asymptotic nature of the vibration frequencies when the number of tubes is large. Our investigation shows that classical homogenization of the problem is not sufficient for this purpose. Indeed, our end result proves that the limit spectrum consists of three parts: the macro-part which comes from homogenization, the micro-part and the boundary layer part. The last two components are new. We describe in detail both macro- and micro-parts using the so-called Bloch wave homogenization method

Homogenization of a Transmission Problem in Solid Mechanics

Artículo de publicación ISIIn this paper we study a simplified model of the behavior of a 3-D solid made from two elastic homogeneous materials separated by a rapidly oscillating interface. We study the asymptotic beha¨ior of the solution of such model using homogenization tools and a compactness result. We obtain the homogenized equation, and by studying its coefficients, we find some properties of the limiting material

Approximate controllability of a semilinear elliptic problem with robin condition in a periodically perforated domain

In this article, we study the approximate controllability and home-genization results of a semi-linear elliptic problem with Robin boundary condition in a periodically perforated domain. We prove the existence of minimal norm control using Lions constructive approach, which is based on Fenchel-Rockafeller duality theory, and by means of Zuazua's fixed point arguments. Then, as the homogenization parameter goes to zero, we link the limit of the optimal controls ( the limit of fixed point of the controllability problems) with the optimal control of the corresponding homogenized problem.SERB-DST Bhopal, YSS-2014-000732 / CREST, IISER Bhopal / CeBiB, PFBasal-01 / CMM, PFBasal-03 / Fondecyt, 1140773 / Regional Program STIC-AmSud Project Mosco

Numerical Maximization of the p-Laplacian Energy of a Two-Phase Material

For a diffusion problem modeled by thep-Laplacian operator, we are interested inobtaining numerically the two-phase material which maximizes the internal energy. We assume thatthe amount of the best material is limited. In the framework of a relaxed formulation, we presenttwo algorithms, a feasible directions method and an alternating minimization method. We show theconvergence for both of them, and we provide an estimate for the error. Since forp >2 both methodsare only well-defined for a finite-dimensional approximation, we also study the difference betweensolving the finite-dimensional and the infinite-dimensional problems. Although the error bounds forboth methods are similar, numerical experiments show that the alternating minimization methodworks better than the feasible directions one

Approximation of solutions to fractional integral equation

In this paper we shall study a fractional integral equation in an arbitrary Banach space X. We used the analytic semigroups theory of linear operators and the fixed point method to establish the existence and uniqueness of solutions of the given problem.Wealso prove the existence of global solution. The existence and convergence of the Faedo Galerkin solution to the given problem is also proved in a separable Hilbert space with some additional assumptions on the operator A. Finally we give an example to illustrate the applications of the abstract results.The first two authors would like to thank the CMM Santiago, University of Chile, for providing the financial support to carry out this research work

A novel model for biofilm growth and its resolution by using the hybrid immersed interface-level set method

Artículo de publicación ISIIn this work we propose a new model to simulate biofilm structures (‘‘finger-like’’, as well as, compact structures) as a result of microbial growth in different environmental conditions. At the same time, the numerical method that we use in order to carry out the computational simulations is new to the biological community, as far as we know. The use of our model sheds light on the biological process of biofilm formation since it simulates some central issues of biofilm growth: the pattern formation of heterogeneous structures, such as finger-like structures, in a substrate-transport-limited regime, and the formation of more compact structures, in a growth-limited-regime. The main advantage of our approach is that we consider several of the most relevant aspects of biofilm modeling, particularly, the existence and evolution of a biofilm–liquid interface. At the same time, in order to perform numerical simulations, we have used sophisticated numerical techniques based on mixing the immersed interface method and the level-set method, which are well described in the present work.This work was partially supported by Millennium Scientific Initiative under grant number ICM P05-001-F. The work of first author was also partially supported by Chilean Government Fondecyt-Conicyt Program under grant number 11080222 and by Universidad del Bío-Bío (grant DIUBB 121909 GI/C). Third author was also partially supported by Fondecyt grant number 1130317

The Bloch Approximation in Periodically Perforated Media

We consider a periodically heterogeneous and perforated medium filling an open domain of RN . Assuming that the size of the periodicity of the structure and of the holes is O(ε), we study the asymptotic behavior, as ε → 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in ε ( ε being minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an approximation of the solution in the energy norm which can be computed from the homogenized solution and the first Bloch eigenfunction.We first consider the case where is RN and then localize the problem for a bounded domain , considering a homogeneous Dirichlet condition on the boundary of

Detecting a moving obstacle in an ideal fluid by a boundary measurement

In this Note we investigate the problem of the detection of a moving obstacle in a perfect fluid occupying a bounded domain in R-2 from the measurement of the velocity of the fluid on one part of the boundary. We show that when the obstacle is a ball, we may identify the position and the velocity of its center of mass from a single boundary measurement. Linear stability estimates are also established by using shape differentiation techniques.This work was achieved while the last author (LR) was visiting the Centro de Modelamiento Matemático at the Universidad de Chile (UMI CNRS 2807). He thanks this institution for its hospitality, and the CNRS for its support. The first author thanks the Millennium ICDB for partial support through grant ICM P05-001-F. The second author was partially supported by CONICYT-FONDECYT grant 3070040. The authors also thank the Chilean and French Governments through Ecos-Conicyt Grant C07 E05

On the graphene Hamiltonian operator

We solve a second-order elliptic equation with quasi-periodic boundary conditions defined on a honeycomb lattice that represents the arrangement of carbon atoms in graphene. Our results generalize those found by Kuchment and Post (Commun Math Phys 275(3):805-826, 2007) to characterize not only the stability but also the instability intervals of the solutions. This characterization is obtained from the solutions of the energy eigenvalue problem given by the lattice Hamiltonian. We employ tools of the one-dimensional Floquet theory and specify under which conditions the one-dimensional theory is applicable to the structure of graphene. The systematic study of such stability and instability regions provides a tool to understand the propagation properties and behavior of the electrons wavefunction in a hexagonal lattice, a key problem in graphene-based technologies.PFBasal-01 (CeBiB) PFBasal-03 (CMM) Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) C13E05 Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 1140773 1180781 Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) 21110749 Spanish Government SEV-2011-008